On a linear refinement of the Pr\'ekopa-Leindler inequality

TL;DR

This paper introduces a linear refinement of the Prékopa-Leindler inequality, based on the common projection of functions onto a hyperplane, extending the inequality's applicability under new geometric assumptions.

Contribution

It establishes a novel linear refinement of the Prékopa-Leindler inequality using the condition of shared projections onto a hyperplane, expanding the inequality's scope.

Findings

Linear refinement of Prékopa-Leindler inequality proved

Refinement based on common hyperplane projection

Extension to Borell-Brascamp-Lieb inequality

Abstract

If $f,g:\mathbb{R}^n\longrightarrow\mathbb{R}_{\geq0}$ are non-negative measurable functions, then the Pr\'ekopa-Leindler inequality asserts that the integral of the Asplund sum (provided that it is measurable) is greater or equal than the $0$-mean of the integrals of $f$ and $g$. In this paper we prove that under the sole assumption that $f$ and $g$ have a common projection onto a hyperplane, the Pr\'ekopa-Leindler inequality admits a linear refinement. Moreover, the same inequality can be obtained when assuming that both projections (not necessarily equal as functions) have the same integral. An analogous approach may be also carried out for the so-called Borell-Brascamp-Lieb inequality.